GENERALISED MULTIVARIATE MIXTURE REGRESSION ESTIMATORS FOR THE POPULATION MEAN WITH MULTI – AUXILIARY CHARACTERISTICS IN MULTI-PHASE SAMPLING

Sat, 01 Jan 2022 00:00:00 GMT

GENERALISED MULTIVARIATE MIXTURE REGRESSION ESTIMATORS FOR THE POPULATION MEAN WITH MULTI – AUXILIARY CHARACTERISTICS IN MULTI-PHASE SAMPLING OLOGUNLEKO, EMMANUEL FEMI Generalised Multivariate Regression Estimators (GMREs) with multi-auxiliary quantitative variables in multi-phase sampling have been used over time to estimate the population mean. These estimators are structurally complex and maximised multiauxiliary quantitative variables only, to produce minimum Mean Square Errors (MSEs). The minimum MSEs can be further reduced with the inclusion of multiauxiliary qualitative variables. However, the existing estimators do not accommodate multi-auxiliary qualitative variables. Therefore, this study was designed to improve the efficiency of the estimators with multi-auxiliary characteristics in multi-phase sampling and simplifying the structurally complex estimators. A population of 𝑁 units, having 𝑌1, 𝑌2, … , 𝑌𝑝 study variables, with 𝑋1, 𝑋2, … , 𝑋𝑡 auxiliary variables and 𝑃1, 𝑃2, … , 𝑃𝑞 auxiliary attributes was considered. The 𝑛ℎ and 𝑛𝑘 (𝑛𝑘 < 𝑛ℎ) are the sample sizes of the ℎ𝑡ℎ and 𝑘𝑡ℎ phases, respectively. Different auxiliary attributes and variables were introduced to the generalised multivariate mixture regression which included Full Information Case (FIC), No Information Case (NIC), Partial Information Case-I (PIC-I), Partial Information Case-II (PIC-II) and Partial Information Case-III (PIC-III). The Improved Estimator Schema (IES) was introduced for the five estimators, in order to simplify the structurally complex estimators. The analytical comparison of the MSEs in five sampling phases was used for the computation of the Percentage Relative Efficiency (PRE) of the estimators. Random deviates of size 𝑁 = 10000 following normal distribution were used to study the behaviour of the estimators asymptotically. Five samples of sizes: 𝑛1, 𝑛2, 𝑛3, 𝑛4 and 𝑛5, with intervals ( 1233 ≤ 𝑛1 ≤ 3333), (542 ≤ 𝑛2 ≤ 1667), ( 361 ≤ 𝑛3 ≤ 1111), ( 271 ≤ 𝑛4 ≤ 833) and (45 ≤ 𝑛5 ≤ 139), were considered for the simulated populations, respectively. The estimators obtained for FIC, NIC, PIC-I, PIC-II, and PIC-III were 𝑡39(1×𝑝), 𝑡40(1×𝑝), 𝑡41(1×𝑝), 𝑡42(1×𝑝) and 𝑡43(1×𝑝), respectively, which were the estimated population means for the multivariate mixture regression estimators in multi-phase sampling with (1 × 𝑝) dimensions. The existing GMREs produced three estimators, which were 𝑡36(1×𝑝), 𝑡37(1×𝑝) and 𝑡38(1×𝑝). The IES obtained for FIC, NIC, PIC-I, PIC-II, and PIC-III estimators which simplified the structurally complex estimators for the multivariate mixture regression estimators in multi-phase sampling were 𝛾𝑡39(1×𝑝),vi 𝛾𝑡40(1×𝑝), 𝛾𝑡41(1×𝑝), 𝛾𝑡42(1×𝑝) and 𝛾𝑡43(1×𝑝), respectively. The corresponding minimised MSEs for the estimators were 𝑀𝑆𝐸(𝑡39)𝑚𝑖𝑛 = 1.9556327, 𝑀𝑆𝐸(𝑡40)𝑚𝑖𝑛 = 2.2219481, 𝑀𝑆𝐸(𝑡41)𝑚𝑖𝑛 = 2.0966104, 𝑀𝑆𝐸(𝑡42)𝑚𝑖𝑛 = 2.1493192 and 𝑀𝑆𝐸(𝑡43)𝑚𝑖𝑛 = 2.2049730, while the corresponding minimised MSEs for the existing estimators were 𝑀𝑆𝐸(𝑡36)𝑚𝑖𝑛 = 1.9714285, 𝑀𝑆𝐸(𝑡37)𝑚𝑖𝑛 = 2.3846115 and 𝑀𝑆𝐸(𝑡38)𝑚𝑖𝑛 = 2.2130263. The proposed estimators have 100.8%, 105.2%, 105.5%, 102.9%, and 100.3% PRE values over the existing estimators, indicating that the proposed estimators were more efficient than the existing estimators. The FIC estimator was the most efficient estimator, while the NIC estimator was the least efficient estimator. Among the partial information case estimators, the PIC-I estimator was conditionally more efficient than PIC-II estimator and PIC-III estimator, while the PIC-II estimator was more efficient than PIC-III estimator. It was observed that the proposed FIC, NIC, PIC-I, PIC-II and PIC-III estimators were asymptotically more efficient. The developed generalised multivariate mixture regression estimators with multiauxiliary characteristics in multi-phase sampling were more efficient in the estimation of the population mean. The structurally complex estimators were simplified by the improved estimator schema.